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4152 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 51, NO. 12, DECEMBER 2005

On the Achievable Diversity–Multiplexing Tradeoffin Half-Duplex Cooperative Channels

Kambiz Azarian, Hesham El Gamal, Senior Member, IEEE, and Philip Schniter, Member, IEEE

Abstract—We propose novel cooperative transmission proto-cols for delay-limited coherent fading channels consisting of(half-duplex and single-antenna) partners and one cell site. In ourwork, we differentiate between the relay, cooperative broadcast(down-link), and cooperative multiple-access (CMA) (up-link)channels. The proposed protocols are evaluated using Zheng–Tsediversity–multiplexing tradeoff. For the relay channel, we inves-tigate two classes of cooperation schemes; namely, amplify andforward (AF) protocols and decode and forward (DF) protocols.For the first class, we establish an upper bound on the achievable di-versity–multiplexing tradeoff with a single relay. We then constructa new AF protocol that achieves this upper bound. The proposedalgorithm is then extended to the general case with ( 1) relayswhere it is shown to outperform the space–time coded protocolof Laneman and Wornell without requiring decoding/encodingat the relays. For the class of DF protocols, we develop a dynamicdecode and forward (DDF) protocol that achieves the optimaltradeoff for multiplexing gains 0 1 . Furthermore,with a single relay, the DDF protocol is shown to dominate the classof AF protocols for all multiplexing gains. The superiority of theDDF protocol is shown to be more significant in the cooperativebroadcast channel. The situation is reversed in the CMA channelwhere we propose a new AF protocol that achieves the optimaltradeoff for all multiplexing gains. A distinguishing feature of theproposed protocols in the three scenarios is that they do not rely onorthogonal subspaces, allowing for a more efficient use of resources.In fact, using our results one can argue that the suboptimality ofpreviously proposed protocols stems from their use of orthogonalsubspaces rather than the half-duplex constraint.

Index Terms—Cooperative diversity, diversity–multiplexingtradeoff, dynamic decode and forward (DDF), half-duplex node,multiple-access channel, nonorthogonal amplify and forward(NAF), relay channel.

I. INTRODUCTION

RECENTLY, there has been a growing interest in the designand analysis of wireless cooperative transmission proto-cols (e.g., [1]–[17]). These works consider several interestingscenarios (e.g., fading versus additive white Gaussian noise(AWGN) channels, ergodic versus quasi-static channels, andfull-duplex versus half-duplex transmission) and devise appro-priate transmission techniques and analysis tools, based on the

Manuscript received July 16, 2004; revised June 30, 2005. This work was sup-ported in part by the National Science Foundation under Grant ITR-0219892.The material in this paper was presented in part at the 38th Annual Conferenceon Information Sciences and Systems, Princeton, NJ, March 2004, the 42ndAllerton Conference on Communication, Control, and Computing, Monticello,IL, October 2004, and the IEEE Information Theory Workshop, San Antonio,TX, October 2004.

The authors are with the Department of Electrical and Computer Engi-neering, The Ohio State University, Columbus, OH 43210-1272 USA (e-mail:[email protected]; [email protected]; [email protected]).

Communicated by R. R. Müller, Associate Editor for Communications.Digital Object Identifier 10.1109/TIT.2005.858920

settings. Here, we focus on the delay-limited coherent channeland adopt the same setup as considered by Laneman, Tse, andWornell in [3]. There, the authors imposed the half-duplexconstraint (either transmit or receive, but not both) on the co-operating nodes and proposed several cooperative transmissionprotocols. In this setup, the basic idea is to leverage the antennasavailable at the other nodes in the network as a source of virtualspatial diversity. The proposed protocols in [3] were classifiedas either amplify and forward (AF), where the helping node re-transmits a scaled version of its soft observation, or decode andforward (DF), where the helping node attempts first to decodethe information stream and then re-encodes it using (a possiblydifferent) codebook. All the proposed schemes in [3] used atime-division multiple-access (TDMA) strategy, where the twopartners relied on the use of orthogonal subspaces to repeateach other’s signals. Later, Laneman and Wornell extendedtheir DF strategy to the partners scenario [4]. Other followupworks have focused on developing practical coding schemesthat attempt to exploit the promised information-theoretic gains(e.g., [5], [6]).

As observed in [3], [4], previously proposed cooperationprotocols suffer from a significant loss of performance in highspectral efficiency scenarios. In fact, the authors of [3] posedthe following open problem: “a key area of further research isexploring cooperative diversity protocols in the high spectralefficiency regime.” This remark motivates our work here, wherewe present more efficient (and in some cases optimal) AF andDF protocols for the relay, cooperative broadcast (CB), andcooperative multiple-access (CMA) channels. To establish thegain offered by the proposed protocols, we adopt the diversity–multiplexing tradeoff as our measure of performance. Thispowerful tool was introduced by Zheng and Tse for point-to-point multiple-input multiple-output (MIMO) channels in [18]and later used by Tse, Viswanath, and Zheng to study the(noncooperative) multiple-access channel in [19].

In the following, we summarize the main results of this paper,some of which were initially reported in [20]–[24].

1. For the single-relay channel, we establish an upper boundon the achievable diversity–multiplexing tradeoff by theclass of AF protocols. We then identify a variant withinthis class, referred to as the nonorthogonal amplify andforward (NAF) protocol, that achieves this upper bound.We then propose a dynamic decode and forward (DDF)protocol and show that it achieves the optimal tradeofffor multiplexing gains .1 Furthermore, theDDF protocol is shown to outperform all AF protocols

1The multiplexing gain “r” will be defined rigorously in the sequel.

0018-9448/$20.00 © 2005 IEEE

AZARIAN et al.: ACHIEVABLE DIVERSITY–MULTIPLEXING TRADEOFF 4153

for arbitrary multiplexing gains. Finally, the two proto-cols (i.e., NAF and DDF) are extended to the scenariowith relays where we characterize their tradeoffcurves. Notably, the NAF protocol is shown to outperformthe space–time-coded protocol of Laneman and Wornell(LW-STC) [4] without requiring decoding/encoding at therelays.

2. For the cooperative broadcast channel, we present a mod-ified version of the DDF protocol to allow for reliabletransmission of the common information. We then char-acterize the tradeoff curve of this protocol and use thischaracterization to establish its superiority compared toAF protocols. In fact, we argue that the gain offered bythe DDF is more significant in this scenario (as comparedto the relay channel).

3. For the symmetric multiple-access scenario, we proposea novel AF cooperative protocol where an artificial inter-symbol-interference (ISI) channel is created. We provethe optimality (in the sense of diversity–multiplexingtradeoff) of this protocol by showing that, for all multi-plexing gains (i.e., ), it achieves the diversity–multiplexing tradeoff of the corresponding point-to-point channel. One can then use this result to arguethat the suboptimality of the schemes proposed in [3] wasdictated by the use of orthogonal subspaces rather thanthe half-duplex constraint. We also utilize this result toshed more light on the fundamental difference betweenhalf-duplex relay and CMA channels.

Before proceeding further, a brief remark regarding two in-dependent prior works [8], [9] is in order. In [7], [8], Nabar,Bölcskei, and Kneubuhler considered the half-duplex single-relay channel, under almost the same assumptions as in [3] (i.e.,the only difference is that, for diversity analysis, the relay-des-tination channel was assumed to be nonfading) and proposeda set of AF and DF protocols. In one of their AF protocols(NBK-AF), Nabar et al. allowed the source to continue trans-mission over the whole duration of the codeword, while the relaylistened to the source for the first half of the codeword and re-layed the received signal over the second half. This makes theNBK-AF protocol identical to the NAF protocol [7], [21]. Here,we characterize the diversity–multiplexing tradeoff achieved bythis protocol while relaxing the assumption of nonfading relay-destination channel which is invoked in the analysis reportedin [8]. Using this analysis, we establish the optimality of thisscheme within the class of linear AF protocols. Furthermore,we generalize the NAF protocol to the case of arbitrary numberof relays and characterize its achieved tradeoff curve. In [9],Prasad and Varanasi derived upper bounds on the diversity–mul-tiplexing tradeoffs achieved by the DF protocols proposed in [8].In the sequel, we establish the gain offered by the proposed DDFprotocol by comparing its diversity–multiplexing tradeoff withthe upper bounds in [9]. Finally, we emphasize that, except forthe single-relay NAF protocol, all the other protocols proposedin this paper are novel.

In this paper, we use to mean , to mean, and to mean nearest integer to toward plus in-

finity. and denote the set of real and complex -tuples,respectively, while denotes the set of nonnegative -tu-

ples. We denote the complement of set , in , by ,while means . denotes the identitymatrix, denotes the autocovariance matrix of vector , and

denotes the base- logarithm.The rest of the paper is organized as follows. In Section II,

we detail our modeling assumptions and review, briefly, someresults that will be extensively used in the sequel. The half-du-plex relay channel is investigated in Section III where we de-scribe the NAF and DDF protocols and derive their tradeoffcurves. In Section IV, we extend the DDF protocol to the co-operative broadcast channel. Section V is devoted to the CMAchannel where we propose a new AF protocol and establish itsoptimality, in the symmetric scenario, with respect to the diver-sity–multiplexing tradeoff. In Section VI, we present numericalresults that show the signal-to-noise (SNR) gains offered by theproposed schemes in certain representative scenarios. Finally,we offer some concluding remarks in Section VII. To enhancethe flow of the paper, we collect all the proofs in the Appendix.

II. BACKGROUND

First, we state the general assumptions that apply to the threescenarios considered in this paper (i.e., relay, broadcast, andmultiple-access). Assumptions pertaining to a specific scenariowill be given in the related section.

1. All channels are assumed to be flat Rayleigh-fading andquasi-static, i.e., the channel gains remain constant duringa coherence interval and change independently from onecoherence interval to another. Furthermore, the channelgains are mutually independent with unit variance. Theadditive noises at different nodes are zero-mean, mutu-ally independent, circularly symmetric, and white com-plex Gaussian. Furthermore, the variances of these noisesare proportional to one another such that there will alwaysbe fixed offsets between the different channels’ s SNRs.

2. All nodes have the same power constraint, have a singleantenna, and operate synchronously. Only the receivingnode of any link knows the channel gain; no feedbackto the transmitting node is permitted (the incremental re-laying protocol proposed in [3] cannot, therefore, be con-sidered in our framework). Following in the footsteps of[3], all cooperating partners operate in the half-duplexmode, i.e., at any point in time, a node can either transmitor receive, but not both. This constraint is motivated by,e.g., the typically large difference between the incomingand outgoing signal power levels. Though this half-duplexconstraint is quite restrictive to protocol development, itis nevertheless assumed throughout the paper.

3. Throughout the paper, we assume the use of randomGaussian codebooks, where a codeword spans the entirecoherence interval of the channel. Furthermore, we as-sume asymptotically large code lengthes. This impliesthat the diversity–multiplexing tradeoffs derived in thispaper serve as upper bounds for the performance of theproposed protocols with finite code lengths. Results re-lated to the design of practical coding/decoding schemesthat approach the fundamental limits established herewill be reported elsewhere.


Next we summarize several important definitions and resultsthat will be used throughout the paper.

1. The SNR of a link is defined as

(1)

where denotes the average energy available for trans-mission of a symbol across the link and denotes thevariance of the noise observed at the receiving end of thelink. We say that is exponentially equal to , de-noted by , when

(2)

In (2), is called the exponential order of . andare defined similarly.

2. Consider a family of codes indexed by operatingSNR , such that the code has a rate of bitsper channel use (BPCU) and a maximum-likelihood (ML)error probability . For this family, the multiplexinggain “ ” and the diversity gain “ ” are defined as

(3)

3. The problem of characterizing the optimal tradeoff be-tween the reliability and throughput of a point-to-pointcommunication system over a coherent quasi-static flatRayleigh-fading channel was posed and solved by Zhengand Tse in [18]. For a MIMO communication system with

transmit and receive antennas, they showed that, forany , the optimal diversity gain isgiven by the piecewise-linear function joining thepairs for ,provided that the code length satisfies .

4. We say that protocol uniformly dominates protocolif, for any multiplexing gain , .

5. Assume that is a Gaussian random variable with zeromean and unit variance. If denotes the exponential orderof , i.e.,

(4)

then the probability density function (pdf) of can beshown to be

Careful examination of the previous expression revealsthat

forfor .

(5)

Thus, for independent random variablesdistributed identically to , the probability that

belongs to set can be characterized by

for (6)

provided that is not empty. In other words, the expo-nential order of only depends on . This is due tothe fact that the probability of any set, consisting of -tu-ples with at least one negative element, de-creases exponentially with SNR and therefore can be ne-glected compared to which decreases polynomiallywith SNR.

6. Consider a coherent linear Gaussian channel, i.e.,

where and denote the signal and noisecomponents of the observed vector, respectively. For thischannel, the pairwise error probability (PEP) of the MLdecoder, denoted as , averaged over the ensemble ofrandom Gaussian codes, is upper bounded by

(7)

7. The following lemma will be used in characterizing thediversity–multiplexing tradeoff of the DDF protocols.

Lemma 1: Consider a coherent linear Gaussian channel ofdata rate and codeword length . The error probability of theML decoder which utilizes a fraction of the codeword such thatthe mutual information between the received and transmittedsignals exceeds , averaged over the ensemble of randomGaussian codes, can be made arbitrarily small provided that thecodeword length is sufficiently large.

Proof: Please refer to the Appendix.

III. THE HALF-DUPLEX RELAY CHANNEL

In this section, we consider the relay scenario in whichrelays help a single source to better transmit its message tothe destination. As the vague descriptions “help” and “bettertransmit” suggest, the general relay problem is rather broad andonly certain subproblems have been studied (for example, see[25]). In this work, we focus on two important classes of relayprotocols. The first is the class of AF protocols, where a re-laying node can only process the observed signal linearly be-fore retransmitting it. The second is the class of DF protocols,where the relays are allowed to decode and re-encode the mes-sage using (a possibly different) codebook. Here we emphasizethat, a priori, it is not clear which class (i.e., AF or DF) offers abetter performance (e.g., [3]).

A. Amplify and Forward (AF) Protocols

We first consider the single-relay scenario (i.e., ).For this scenario, we derive the optimal diversity–multiplexingtradeoff and identify a specific protocol within this class, i.e.,the NAF protocol [7], [21], that achieves this optimal tradeoff.


We then extend the NAF protocol to the general case with anarbitrary number of relays.

Under the half-duplex constraint, it is easy to see that anysingle-relay AF protocol can be mathematically described bysome choice of the matrices , , and in the followingmodel:

(8)

In (8), represents the vector of observations at thedestination, the vector of source symbols,the vector of noise samples (of variance ) observed by therelay, and the vector of noise samples (of variance

) observed by the destination. The variables , , anddenote the source-relay channel gain, source-destination

channel gain, and relay-destination channel gain, respectively.and are diagonal matrices. In

this protocol, the source can potentially transmit a new symbolin every symbol interval of the codeword, while the relaylistens during the first symbols and then, for the remaining

symbols, transmits linear combinations of the noisyobservations using the coefficients in . In fact,by letting , , , and (with

denoting the relay repetition gain),we obtain Laneman–Tse–Wornell AF (LTW-AF) protocol [3].Finally, we note that when the source symbols are independent,the average energy constraint translates to

(9)

where and .

Theorem 2: The optimal diversity gain for the cooperativerelay scenario with a single AF relay is upper-bounded by

(10)


The upper bound on , as given by (10), is shown inFig. 1. Having Theorem 2 at hand, it now suffices to identifyan AF protocol that achieves this upper bound in order to es-tablish its optimality. Toward this end, we observe that, in theproof of Theorem 2, the only requirements on such that theprotocol described by (8) could potentially achieve the optimaldiversity–multiplexing tradeoff are for to be square (of di-mension ) and full-rank. Furthermore, should notviolate the relay average energy constraint as given by (9). Thus,the simple choices

for (11)

inspire the NAF protocol [7], [21]. In particular, the sourcetransmits on every symbol interval in a cooperation frame,where a cooperation frame is defined as two consecutive

Fig. 1. Optimal diversity–multiplexing tradeoff for a single-relay AF protocol.

symbol intervals. The relay, on the other hand, transmits onlyonce per cooperation frame; it simply repeats the (noisy) signalit observed during the previous symbol interval. It is importantto realize that this design is dictated by the half-duplex con-straint, which implies that the relay can repeat at most once percooperation frame. We denote the repetition gain by and, forframe , we denote the information symbols by . Thesignals received by the destination during frame are thus,

where the repetition gain must satisfy (11). Note that, inorder to decode the message, the destination needs to knowthe relay repetition gain , the source-relay channel gain , thesource-destination channel gain , and the relay-destinationchannel gain . Now, we are ready to establish the optimalityof the NAF protocol with respect to the diversity–multiplexingtradeoff.

Theorem 3: The NAF protocol achieves the optimal diver-sity–multiplexing tradeoff for the AF single-relay scenario,which is

(12)

Proof: Please refer to the Appendix.Three remarks are now in order.

1. As shown in Fig. 1, the NAF protocol enjoys uniformdominance over the direct transmission scheme (i.e., nocooperation) and LTW-AF protocol. This dominancecan be attributed to relaxing the orthogonality constraintwhereby one can reap two distinct benefits: rate en-hancement via continuous transmission and diversityenhancement via cooperation. It is interesting to notethat this dominance is achieved while only half of thesymbols are repeated by the relay.

2. From Fig. 1, one can see that for multiplexing gainsgreater that , the diversity gain achieved by the NAFrelay protocol is identical to that of the noncooperative


Fig. 2. The super-frame in the NAF protocol with N � 1 relays.

protocol. This is due to the fact that the AF cooperativelink provided by the relay cannot support multiplexinggains greater than —a consequence of the half-duplexconstraint. Hence, for multiplexing gains larger than ,there is only one link from the source to the destination,and, thus, the tradeoff curve is identical to that of apoint-to-point system with one transmit and one receiveantenna. Later, we will show that the proposed DDFstrategy avoids this drawback.

3. As shown in the proof of Theorem 3, the achievability ofthe optimal tradeoff is not very sensitive to the choice ofthe repetition gain “ ” (i.e., for a wide range of choices,the NAF protocol achieves the optimal tradeoff). In prac-tice, one should optimize the repetition gain, experimen-tally if needed, to minimize the outage probability at thetarget rate and SNR.

The NAF protocol can be extended to the case of arbitrarynumber of relays (i.e., ) as follows. First, we define asuper-frame as a concatenation of consecutive cooper-ation frames. Within each super-frame, the relays take turnsrepeating the signals they previously observed as they did in thecase of a single relay (refer to Fig. 2). Thus, the destination’sreceived signals during a super-frame will be

...

where the source-relay channel gain, relay-destination channelgain, relay-repetition gain, and relay-observed noise for relay

are denoted by , , , and ,respectively. As before, represents the source–destinationchannel gain. The quantities , , and represent thereceived signal, noise sample, and source symbol, respectively,during the th symbol interval of the th cooperation frame. Notethat there is nothing to be gained by having more than one relaytransmitting the same symbol simultaneously. Also, similar tothe single-relay NAF scenario, the destination needs to knowall relay repetition gains as well as all channel gains

and . The following theorem characterizes thediversity–multiplexing tradeoff achieved by this protocol.

Theorem 4: The diversity–multiplexing tradeoff achieved bythe NAF protocol with relays is characterized by

Proof: The proof is virtually identical to that of The-orem 3, and hence, is omitted for brevity.

It is interesting to note that the generalized NAF protocol uni-formly dominates the LW-STC. This can be attributed to thefact that in the generalized NAF protocol, in contrast to theLW-STC protocol, the source transmits over the whole durationof the codeword. The generalized NAF protocol offers the ad-ditional advantage of low complexity since it does not requiredecoding/encoding at the relays.

B. Decode and Forward (DF) Protocols

In this class of protocols, we allow for the possibility ofdecoding/encoding at the different relays. In [3], Laneman,Tse, and Wornell presented a particular variant of DF protocols(LTW-DF) where the source transmits in the first half of thecodeword. Based on its received signal in this interval, the relayattempts to decode the message. It then re-encodes and trans-mits the encoded stream in the second half of the codeword. In[4], Laneman and Wornell derived the diversity–multiplexingtradeoff achieved by this scheme (i.e., ), whichis depicted in Fig. 3. Here, we propose a DDF protocol andcharacterize its tradeoff curve. This characterization revealsthe uniform dominance of this protocol over all known full-diversity (i.e., ) protocols proposed for the half-duplexsingle-relay channel and furthermore establishes its optimality,over a certain range of multiplexing gains (i.e., ).We first describe and analyze the protocol for the case of asingle relay. Generalization to relays will then follow.

Similar to the previous section, we assume that a codewordconsists of consecutive symbol intervals, during which allthe channel gains remain unchanged. In the DDF protocol,the source transmits data at a rate of BPCU during everysymbol interval in the codeword. The relay, on the other hand,listens to the source until the mutual information between itsreceived signal and source signal exceeds . It then decodesand re-encodes the message using an independent Gaussiancodebook and transmits it during the rest of the codeword. The


Fig. 3. Diversity–multiplexing tradeoff for the DDF protocol with one relay.

dynamic nature of the protocol is manifested in the fact that weallow the relay to listen for a time duration that depends on theinstantaneous channel realization to maximize the probabilityof successful decoding. We denote the signals transmitted bythe source and relay as and , respectively,where is the number of symbol intervals the relay waitsbefore starting transmission. Using this notation, the receivedsignals (at the destination) can be written as

for

for

From the protocol description, it is clear that the number of sym-bols where the relay listens should be chosen as

(13)

where is the source-relay channel gain, and . Onecan now see the dependence of this choice of on the instanta-neous channel realization and that this choice, together with theasymptotically large , guarantees that when , the relayaverage probability of error with a Gaussian code ensemble isarbitrarily small.2 Clearly, when , the relay does not con-tribute to the transmission of the message, and hence, incorrectdecoding at the relay in this case does not affect performance.Here, we observe that, in contrast to the NAF protocol, the des-tination does not need to know the source-relay channel gain.It does, however, need to know the relay waiting time , along,with the source-destination and relay-destination channel gains.The following theorem describes the diversity–multiplexingtradeoff achievable with this cooperation protocol.

Theorem 5: The diversity–multiplexing tradeoff achieved bythe single-relay DDF protocol is given by

ifif .

(14)


2This point will be established rigorously in the proofs of Theorem 5 andLemma 1.

The diversity–multiplexing tradeoff of (14) is shown in Fig. 3.It is now clear that the DDF protocol is optimal forsince it achieves the genie-aided diversity (where the relay isassumed to know the information message a priori). For

, the DDF protocol suffers from a loss, compared to thegenie-aided strategy, since, on the average, the relay will onlybe able to help during a small fraction of the codeword. It iseasy to see that, the performance for this range of multiplexinggains cannot be improved through employing a mixed AF andDF strategy. In fact, the DDF strategy dominates all such strate-gies.3 It remains to be seen whether there exists a strategy thatcloses the gap to the genie-aided strategy when or not.Note also that the gain offered by the DDF protocol, comparedto AF protocols, can be attributed to the ability of this strategyto transmit independent Gaussian symbols after successful de-coding. In AF strategies, in contrast, the relay is limited to re-peating the noisy Gaussian symbols it receives from the source.Fig. 3 also compares the DDF protocol with the DF protocol pro-posed in [8], which we refer to as NBK-DF. In this comparison,we utilize the upper bound derived by Prasad and Varanasi onthe diversity–multiplexing tradeoff of the NBK-DF, which wasreported in [9]. One can see from Fig. 3 that the NBK-DF pro-tocol does not achieve any diversity gain greater than one. Thiscan be attributed to the fact that in this protocol, the message issplit up into two parts, out of which, only one is retransmittedby the relay. Fig. 3 also shows that for multiplexing gains closeto one, the NBK-DF upper bound outperforms the DDF pro-tocol. Therefore, in this range, the comparison between the twoprotocols depends on the tightness of the NBK-DF upper boundwhich was not discussed in [9].

Next, we describe the generalization of the DDF protocol tothe case of multiple relays. In this case, the source and relayscooperate in nearly the same manner as in the single-relay case.Specifically, the source transmits during the whole codewordwhile each relay listens until the mutual information betweenits received signal and the signals transmitted by the source andother relays exceeds . It is assumed that every relay knowsthe codebooks used by the source and other relays. Once a relaydecodes the message, it uses an independent codebook to re-en-code the message, which it then transmits for the rest of thecodeword. Note that, since the source-relay channel gains maydiffer, the relays may require different wait times for decoding.This complicates the protocol, since a given relay’s ability todecode the message requires precise knowledge of the timesat which every other relay begins its transmission. To addressthis problem, the codeword is divided into a number of seg-ments, and relays are allowed to start transmission only at thebeginning of a segment. In between the segments, every relayis allowedto broadcast a (well-protected) beacon, informing allother relays whether or not it will start transmission. Judiciouschoice of the segment length, relative to the codeword length,results in only a small loss compared to the genie-aided case,whereby all relays know all decoding times a priori. Here, weassume that the number of segments is sufficiently large and thelength of the beacon signals is much smaller than the segment

3The proof for this is rather straightforward, and hence, is omitted here forbrevity.


Fig. 4. Diversity–multiplexing tradeoff for the NAF, DDF, LW-STC, andgenie-aided protocols with four relays.

Fig. 5. Diversity–multiplexing tradeoff for the DDF protocol with differentnumber of relays.

length. Therefore, in characterizing the diversity–multiplexingtradeoff achieved by this protocol, we ignore the losses associ-ated with the beacons and the quantization of the starting timesfor the different relays.

Theorem 6: The diversity–multiplexing tradeoff achieved bythe DDF protocol with relays is characterized by

.

(15)


The diversity-multiplexing tradeoff (15) is shown in Figs. 4and 5 for different values of . While the loss of the DDF pro-tocol compared to the genie-aided protocol increases with , itis not clear at the moment if this loss is due to the half-duplexconstraint or due to the suboptimality of the DDF strategy.

IV. THE HALF-DUPLEX COOPERATIVE BROADCAST(CB) CHANNEL

We now consider the CB scenario, where a single sourcebroadcasts to destinations. This model corresponds to a gen-eralized version of a single-cell downlink where the destina-tions are allowed to cooperate through helping one another inreceiving their messages. Similar to the relay channel scenario,we adopt the quasi-static flat Raleigh-fading model for all thesource-destination and inter-destination channels. We assumethat the message intended for destination con-sists of two parts. A common part of rate BPCU,which is intended for all of the destinations and an individualpart of rate BPCU, which is specific to the thdestination. The total rate is then and themultiplexing gain tuple is given by . Wedefine the overall diversity gain based on the performance ofthe worst receiver as

where we require all the receivers to decode the common infor-mation.4 Now, as a first step, one can see that if , i.e.,if there is no common message, then the techniques developedfor the relay channel can be exported to this setting through aproportional time sharing strategy. With this assumption, all ofthe properties of the NAF and DDF protocols, established forthe relay channel, carry over to this scenario. The problem be-comes slightly more challenging when . In fact, it is easyto see that, for a fixed total rate , the highest probability oferror corresponds to the case where all destinations are requiredto decode all the messages. This translates to the following con-dition (that applies to any cooperation scheme):

(16)

So, we will focus the following discussion on this worst casescenario, i.e.,

(17)

The first observation is that, in this scenario, the only AFstrategy that achieves the full rate extreme pointis the noncooperative protocol. Any other AF strategy will re-quire some of the nodes to retransmit, and therefore not to listenduring parts of the codeword,5 which prevents it from achievingfull rate. Fortunately, this drawback can be avoided in the DDFprotocol. The reason is that, in this protocol, any node will starthelping only after it has successfully decoded the message. Wenow propose a protocol for the CB scenario that is a direct ex-tension of the DDF relay protocol. This will be referred to as theCB-DDF protocol in the sequel. The only modification needed,compared to the relay channel case, is that now every destinationcan act as a relay for the other destinations, based on its instanta-neous channel gain. Specifically, the source transmits during thewhole codeword while each destination listens until the mutualinformation between its received signal and the signals trans-mitted by the source and other destinations exceeds . Once a

4Clearly, this definition does not allow for different quality of service (QoS)constraints.

5This follows from the half-duplex constraint.


Fig. 6. The cooperation frame, super-frame, and coherence interval in theCMA-NAF protocol with N sources.

destination decodes the message, it uses an independent code-book to re-encode the message, which it then transmits for therest of the codeword. Similar to the relay channel, it is assumedthat every destination knows the codebooks used by the sourceand other destinations. Also, the protocol must include a mech-anism that keeps every destination informed of the retransmis-sion starting times of all the other destinations. Again, in de-riving the following result, we ignore the associated cost of thismechanism, relying on the asymptotic assumptions.

Theorem 7: The diversity–multiplexing tradeoff achieved bythe CB-DDF protocol with destinations is given by

.(18)


It is interesting to note that this is exactly the same tradeoffobtained in the relay channel. This implies that requiring allnodes to decode the message does not entail a price in termsof the achievable tradeoff.

V. THE HALF-DUPLEX COOPERATIVE MULTIPLE-ACCESS(CMA) CHANNEL

In this section, we consider the CMA scenario, wheresources transmit their independent messages to a common des-tination. This model corresponds to a generalized version of asingle-cell uplink where the sources are allowed to cooperate.The CMA scenario was previously considered in [1], [3] (andreferences therein). Again, the same quasi-static flat Rayleigh-fading model is invoked for all the channels. We assume sym-metry so that all sources transmit information at the same rateand are limited by the same power constraint. The basic idea ofthe proposed protocol, which we refer to as the CMA-NAF pro-tocol, is to create an artificial ISI channel. Toward this end, eachof the sources transmits once per cooperation frame, where acooperation frame is defined as consecutive symbol-intervals(refer to part a) of Fig. 6). Each source is assigned unique trans-mission and reception symbol intervals within the cooperation

frame. During its transmission symbol interval, a source trans-mits a linear combination of its own symbol and the signal it ob-served during its most recent reception symbol interval. In otherwords, every source, in addition to sending its own symbol,helps another source by repeating the (noisy) signal it last re-ceived from it. Without loss of generality, we set the th sourcetransmission symbol interval equal to .

We now provide an illustrative example for the case.Here we assume that sources 1, 2, and 3 help sources 3, 1,and 2, respectively. For the th source and the th cooperationframe, denotes the transmission, the (assigned) recep-tion, and the originating symbol. Using and to denotethe broadcast and repetition gains of the th source, respectively,the signals transmitted during the first two cooperation frameswould be (in chronological order)

Using to denote the th-source-to- th-source channel gain,and to denote the noise observed by the th source duringits th-frame reception symbol interval, the assigned receptionsbecome

Using to denote the th-source-to-destination channel gain,and to denote the noise observed by the destination duringthe th symbol interval of the th frame, the signals observed atthe destination would be

The source-observed noises have variance for all , ,and the destination-observed noises have variance forall , . Note that, as mandated by our half-duplex constraint, nosource transmits and receives simultaneously. The broadcast andrepetition gains should be chosen to satisfy the averagepower constraint

(19)

Let us now define consecutive cooperation frames as asuper-frame (refer to part b) of Fig. 6). We will assume thathelper assignments are fixed within a super-frame but are sched-uled to change across super-frames. We impose the followingrequirements on helper scheduling.

1. In each super-frame, every source is helped by a differentsource.

2. Across super-frames, every source is helped equally byevery other source.


Fig. 7. Comparison of the outage probability for the NAF relay, LTW-AF, and noncooperative 1 � 1 protocols (N = 2).

Among the many scheduling rules that satisfy these require-ments, we choose the following circular rule. In super-frame ,sources with indices are assigned helpers with in-dices given by the th right circular shift of , where

. For example, when , the helperconfigurations are given by the following table.

Super-frameindex

Helper assigned to

Since this scheduling algorithm generates distinct helperconfigurations, the length of the super-frames is chosen suchthat a coherence-interval consists of consecutive super-frames (refer to part c) of Fig. 6). To achieve maximal diversityfor a given multiplexing gain, it is required that all codewordsspan the entire coherence interval. For this reason, we choosecodes of length given by

(20)

Similar to the broadcast channel, defining the multiplexinggain and diversity gain for the CMA channel requires somecare. Note that, using (3), the pair can be defined forcommunication between the th source and the destination.However, since we assumed a symmetric CMA setup, allmultiplexing gains are equal, i.e., for all . Furthermore,since CMA-NAF mandates that only one source transmits inany symbol interval, the destination’s multiplexing gain is alsoequal to . That is, the destination receives information at rate

given by

(21)

We define the overall diversity gain based on the worst caseprobability of error for the information streams, i.e.,

With these definitions, Theorem 8 establishes the optimality ofthe CMA-NAF in the symmetric scenario with sources.

Theorem 8: The CMA-NAF protocol achieves the optimal(genie-aided) diversity–multiplexing tradeoff for the symmetricscenario with sources, given by

(22)


Theorem 8 not only establishes the optimality of theCMA-NAF protocol, but also it shows that the half-duplexconstraint does not entail any cost, in terms of diversity–mul-tiplexing tradeoff, in the symmetric CMA channel. One cannow attribute the suboptimality of the CMA schemes reportedin [3], [4] to the use of orthogonal subspaces. It is interestingto observe that one can achieve the optimal tradeoff in thesymmetric CMA channel with a simple AF strategy. In fact,by comparing Theorems 2 and 8, one can see the fundamentaldifference between the half duplex CMA and relay channels.

VI. NUMERICAL RESULTS

In this section, we report numerical results that quantify theperformance gains offered by the proposed protocols. Thesenumerical results correspond to outage probabilities and aremeant to show that the superiority of the proposed protocolsin terms of diversity-multiplexing tradeoff translates into sig-nificant SNR gains. In Figs. 7–9, we compare the proposedprotocols with the noncooperative (direct transmission) and


Fig. 8. Comparison of the outage probability for the DDF relay, LTW-AF, and noncooperative 1 � 1 protocols (N = 2).

Fig. 9. Comparison of the outage probability for the CMA-NAF, LTW-AF, and genie-aided 2 � 1 protocols (N = 2).

the LTW-AF protocols. To ensure fairness, we have imposedmore strict power constraints on the NAF and the DDF relayprotocols; specifically, we lowered the average transmissionenergy of the source and the relay from to during theinterval when both are transmitting. This way, the total averageenergy per symbol interval, spent by any of the protocols con-sidered here6 is . While one may find other energy allocationstrategies that offer performance improvement (in terms of theoutage probability), any such optimization will not affect theachievable diversity–multiplexing tradeoff, and hence, will notbe pursued here. To obtain lower bounds on the gains offeredby the DDF and CMA-NAF protocols, we assume a noiseless

6In the CMA-NAF protocol, the constant average energy per symbol intervalis automatically implied.

source-relay channel for the LTW-AF and NAF relay protocols.For the DDF relay and the CMA-NAF protocols, the SNR ofthe link between the two cooperating partners was assumedto be only 3 dB better than that of the relay-destination orsource-destination channels. We optimized the broadcast andrepetition gains for the CMA-NAF protocol experimentally. Inall the considered cases, the outage probabilities are computedthrough Monte Carlo simulations.

Fig. 7 shows the performance gain offered by the NAFrelay protocol over both the noncooperative protocol andthe LTW-AF protocol at high SNRs and two different datarates. The same comparison is repeated in Fig. 8 with theDDF protocol where, as expected, the gains are shown to belarger. The CMA channel is considered in Fig. 9, where the


optimality of the CMA-NAF protocol is shown to translateinto significant SNR gains. It is also interesting to note thatthe gap between CMA-NAF performance and genie-aidedstrategy is less than 3 dB when the date rate is equal to 2 BPCU.We can also observe that the gains offered by the DDF andCMA-NAF protocols compared with the LTW-AF protocolincrease with the data rate. This is a direct consequence of thehigher multiplexing gains achievable with our newly proposedprotocols. Overall, these results re-emphasize the fact that thefull diversity criterion alone7 is a rather weak design tool.

We conclude this section with a brief comment on our choicefor the diversity–multiplexing tradeoff as our design tool. Thischoice is inspired by the convenient tradeoff, between analyticaltractability and accuracy, that this tool offers. Ideally, one shouldseek cooperation schemes that minimize the outage probabilityat the target rate and SNR. Unfortunately, it is easy to see thatsuch an approach would lead to an intractable problem even invery simplified scenarios. Our results, on the other hand, demon-strate that one can use the diversity–multiplexing tradeoff toanalytically guide the design in many relevant scenarios. Fromthe accuracy point of view, our simulation results validate thatschemes with better tradeoff characteristics always offer signif-icant SNR gains at sufficiently high SNRs. In this context, themain drawback of the diversity–multiplexing tradeoff is that itfails to predict at which SNR the promised gains will start toappear. For example, from the figures, one can see that the DDFand CMA-NAF schemes yield performance gains at relativelymoderate SNRs whereas the NAF protocol only offers gain atlarger SNRs.

VII. CONCLUSION

In this paper, we considered the design of cooperative proto-cols for a system consisting of half-duplex nodes. In particular,we differentiated between three scenarios. For the relay channel,we investigated the AF and DF protocols. We established theuniform dominance of the proposed DDF protocol compared toall known full diversity cooperation strategies and its optimalityin a certain range of multiplexing gains. We then proceeded tothe cooperative broadcast channel where the gain offered by theDDF strategy was argued to be more significant, as comparedto the relay channel. For the multiple-access scenario, we pro-posed a novel AF cooperative protocol where an artificial ISIchannel was created. We proved the optimality (in the sense ofdiversity–multiplexing tradeoff) of this protocol by showing thatit achieves the same tradeoff curve as the genie-aidedpoint-to-point system.

Our results reveal interesting insights on the structure ofoptimal cooperation strategies with half-duplex partners. First,we observe that, without the half-duplex constraint, achievingthe optimal tradeoff in the three channels considered here israther straightforward (i.e., one can easily construct a simpleAF strategy that results in an -tap ISI channel, and hence,the optimal tradeoff). With the half-duplex constraint, morecare is necessary in constructing the cooperation strategies, but,as shown, one can still achieve the optimal tradeoff in manyrelevant scenarios. One of the important insights is that one

7Full diversity corresponds to the point (d = 2; r = 0) on the tradeoff curve.

should strive to transmit independent symbols as frequently aspossible (as observed in [7], [21] for the relay channel case).Indeed, the optimality of the proposed CMA-NAF protocolstems from exploiting the distributed nature of the informationto enable transmission of an independent symbol in everysymbol interval. It is now easy to see that the use of orthogonalsubspaces to enable cooperation, as in [3] for example, entailsa significant loss in the achievable tradeoff.

This work poses many interesting questions. For example,proving (or disproving) the optimality of the DDF protocol forthe single-relay channel and is an open problem. Gener-alizations of the proposed schemes to multiple-antenna nodes,cooperative automatic retransmission request (ARQ) channels[23], scenarios with different QoS constraints, and asymmetricCMA channels are of definite interest. Finally, the design ofpractical coding/decoding strategies that approach the funda-mental limits achievable with Gaussian codes and ML decodingis an important venue to pursue.

APPENDIX

In this appendix, we collect all the proofs.

A. Proof of Lemma 1

For simplicity, we consider the real AWGN channel (the com-plex channel is a straightforward extension), i.e.,

for

where is the received signal over the th symbol interval,is the channel gain, and is the independent and identically

distributed (i.i.d.) Gaussian noise sample during symbol interval. In our analysis, we derive the average probability of error,

where averaging is invoked with respected to the ensemble ofGaussian codes, conditioned on a particular , which is assumedto be known at the decoder. We denote this error probability by

. Toward this end, we divide a codeword into segments,each consisting of symbol intervals, i.e., . In ouranalysis, we consider the asymptotic scenario where andgrow to infinity. The ML decoder waits for segments beforeit starts decoding, where is given by

(23)

In (23), denotes the mutual information betweenand . Notice that assuming the data rate to be less than thechannel capacity guarantees . Also, observe that thefraction of codeword that the decoder has to wait before de-coding is given by

Our analysis is based on exactly the same techniques used toestablish the achievability of the AWGN channel capacity, i.e.,we first upper-bound the ML error probability with that of atypical set decoder and then take the average of the latter errorprobability, over the ensemble of random Gaussian codebooks.


In particular, following exactly the same argument as in [26,Theorem 10.1.1], we get

for sufficiently large and . This follows from the choice ofin (23). It is important to note that the transmission rate

is constant, i.e., independent of , and is not known to thetransmitter.

B. Proof of Theorem 2

Due to the source average energy constraint, setting andto anything other than the identity matrix will reduce the

mutual information between and . Since we are interested inobtaining an upper bound, we will choose and

, in which case (8) reduces to

(24)

Using singular value decomposition (SVD), the matrix canbe factored as

where and are unitary andwhere is nonnegative diagonal with the diag-onal elements in decreasing order. Using these matrices, we de-fine , , , and , for unitarytransformation

The unitary property of and implies that and, as well as

(25)

In terms of the new variables, (24) becomes

(26)

with

(27)

If we denote the nonzero diagonal elements of as ,then (27) can be written as

and

where , , and represent the th element of , , and ,respectively, and where , ,

, and

(28)

(29)

Note that, according to the SVD theorem

(30)

Because is diagonal, (and therefore, ) is maxi-mized when are mutuallyindependent, in which case we would have

(31)

The mutual information between and is given by

(32)

A lower bound on is easily obtained by re-placing by

(33)

Since is an increasing function on the cone of pos-itive-definite Hermitian matrices and since(where represents the largest eigenvalue of ), we getthe following upper bound on :

(34)

From (33) and (34), we conclude that

Now, since is of the same exponential order as , thebounds converge as grows to infinity. That is,


Plugging in for and from (28) and (29), respectively, weget

It is then straightforward to see that

(35)

where , , and are the exponential orders of ,, and , respectively. In deriving this expression,

we have assumed that ; as explained earlier,we do not need to consider realizations in which , , orare negative. Similarly

which, together with (35), (25), and (31), results in

(36)

For the quasi-static fading setup, the outage event is definedas the set of channel realizations for which the instantaneouscapacity falls below the target data rate. Thus, our outage event

becomes

Letting grow with according to

and using (36), we conclude that, for large

(37)

and thus,

(38)

As Zheng and Tse have shown in [18, Lemma 5], pro-vides an upper bound on (i.e., the optimal diversity gainat multiplexing gain )

(39)

From (37) and (38), it is easy to see that the right-hand sideof (39) is maximized when is set to its maximum, which,

according to (30), is . This is the case when isfull-rank. On the other hand, itself is maximizedwhen (assuming an even codeword length ), whichcorresponds to being a square matrix. For this , canbe shown to take the value of the right-hand side of (10). Thiscompletes the proof.

C. Proof of Theorem 3

The proof closely follows that for the MIMO point-to-pointcommunication system in [18]. In particular, we assume that thesource uses a Gaussian random codebook of codeword length ,where is taken to be even, and data rate , where increaseswith according to

The error probability of the ML decoder, , can be upper-bounded using Bayes’ rule

where denotes the outage event. The outage event is chosensuch that dominates , i.e.,

(40)

in which case

(41)

In order to characterize , we note that, since the destinationobservations during different frames are independent, the upperbound on the ML conditional PEP (recalling (7)), assuming tobe even, changes to

(42)

where and denote the covariance matrices of destina-tion observation’s signal and noise components during a singleframe

(43)

(44)

Let us define , , , and as the exponential orders of, , , and , respectively. Then the con-

straint on given in (11) implies the following constraint on :

(45)

We assume is chosen such that the exponential orderbecomes


Fig. 10. Outage region for the NAF protocol with a single relay.

which satisfies the constraint given by (45). Interestingly, if weconsider , then becomes zero and vanishesin the expressions. Plugging (43) and (44) into (42), we obtain

With rate BPCU and codeword length , we have atotal of codewords. Thus,

for

is the average of over the set of channel re-alizations that do not cause an outage (i.e., ). Using (5), onecan see that

for

Now, is dominated by the term corresponding to the min-imum value of over

(46)

Using (6), can be expressed

(47)

Comparing (46) and (47), we realize that for (40) to be met,should be defined as

Then, for any , it is possible to choose tomake arbitrarily large, ensuring (40). Note that,because of (41), provides a lower bound on the diversitygain achieved by the protocol. But , as given by (47), turnsout to be identical to right-hand side of (10) (refer to Fig. 10).Thus, the optimal diversity–multiplexing tradeoff for this sce-nario is indeed given by (12) and the NAF protocol achieves it.

D. Proof of Theorem 5

Instead of considering specific codes, in the following weupper-bound the average probability of error over randomGaussian ensemble of codebooks (employed by both the sourceand relay). Therefore, averaging is invoked with respect to boththe fading channel distribution and the random codebooks. It isthen straightforward to see that there is at least one codebookin this ensemble whose average performance, now with respectonly to the fading channel distribution, is better than the predic-tions of our upper bound. For the single-relay DDF protocol, theerror probability of the ML decoder, averaged over the ensembleof Gaussian codebooks and conditioned on a certain channelrealization, can be upper-bounded using Bayes’ rule to give

where and denote the events that the relay decodessource’s message erroneously and its complement, respec-tively. The first step in the proof follows from Lemma 1 byobserving that if (13) is met, i.e., if the mutual informationbetween the signal transmitted by the source and the signalreceived by the relay exceeds , then can be madearbitrarily small, provided that the code length is sufficientlylarge. This means that for any and for a sufficiently largecode length,

Taking the average over the ensemble of channel realizationsgives

This means that the exponential order of , i.e., destina-tion’s ML error probability assuming error-free decoding at therelay, provides a lower bound on the diversity gain achieved bythe protocol. Therefore, we only need to characterize ,which for the sake of notational simplicity, we will denote by

in the sequel. To characterize , we note that the corre-sponding PEP (recalling (7)) is given by

Defining , , and as the exponential orders of ,, and , respectively, gives

for

where . At a rate of BPCU and a codewordlength of , there are a total of codewords. Thus,

for

(48)


Examining (48), we realize that for (40) to hold, should bedefined as

(49)

so that it is possible to choose to make arbitrarily large,ensuring (40). As before, is given by (47), which turnsout to be identical to given by (14). To see this, one needsto consider four different categories of channel realizations. Thefirst category is when both and are greater than one. Forthis category

(50)

The second category is when and . It is easyto see from (49) that for this category

(51)

The third category to be considered is when and. Before proceeding further, note that from (13), one can

show that

(52)

This implies that , since is nonnegative. Returning backto (49), it is easy to verify that for this category

(53)

Now, if , then from (53) and (52) we get

or

.

(54)On the other hand, if , then

or

for

This means that , for the third category, isindeed given by (54). It is noteworthy that the tradeoff curvesgiven by (50), (51), and (54), are all better than the genie-aided

Fig. 11. Outage region for the DDF protocol with a single relay (f � 0:5).

Fig. 12. Outage region for the DDF protocol with a single relay (f > 0:5).

tradeoff. In other words, the diversity gain achieved by this pro-tocol is determined by the fourth category, where both andare less than or equal to one. For this category, one needs to con-sider two cases (note that (52) is still valid, implying ). Thefirst case, when , is very easy. Referring to Fig. 11reveals that, in this case, and, therefore,

is equal to (the genie-aided tradeoff). The secondcase, when , is a little bit more difficult. As canbe seen from Fig. 12, in this case

(55)

From (52) and (55), we conclude that

(56)

which gives (14). Again, according to (41), provides alower bound on the diversity gain achieved by the protocol. Onthe other hand, is also an upper bound on the achieved


diversity since: 1) for is the genie-aideddiversity and 2) for , it is easy to see that

, , and correspond to a channel outagefor any . Thus, (14) is the diversity achieved by the DDFprotocol and the proof is complete.

E. Proof of Theorem 6

Inspired by the single-relay case, we use ensembles ofGaussian codebooks at the source and all the relays. To charac-terize the diversity–multiplexing tradeoff achieved by the DDFprotocol with relays, we first label the nodes accordingto the order in which they start transmission. That is, the sourceis labeled as node 1, the first relay that starts transmission asnode 2, and so on. We then use Bayes’ rule to upper-boundthe error probability of the ML decoder, averaged over theensemble of Gaussian codebooks and conditioned on a certainchannel realization, to get

(57)

where , , denotes the event that node de-codes the source message in error, while denotes its com-plement. Let us now examine , i.e., the prob-ability that node makes an error in decodingthe source message, assuming error-free decoding at all pre-vious nodes. It follows from Lemma 1, that if the mutual in-formation between the signals transmitted by the source and ac-tive relays and the signal received by node exceeds , then

can be made arbitrarily small, provided thatthe code length is sufficiently large. This means that for any

and for sufficiently large code lengths

(58)

Using (58), (57) can be written as


This means that the exponential order of , i.e., des-tination’s ML error probability assuming error-free decoding atall of the relays, provides a lower bound on the diversity gainachieved by the protocol. Therefore, we only need to charac-terize , which for the sake of notational simplicity,we will denote by in the sequel. To characterize , we notethat the corresponding PEP is upper-bounded by

As before, the gain of the channel that connects the th node tothe destination is denoted by , while the gain of the channelthat connects nodes and is denoted by . We use todenote the number of symbol intervals in the codeword duringwhich a total of nodes are transmitting, so that ,with denoting the total codeword length. Note thatis the number of symbol intervals that relay has to wait,before the mutual information between its received signal andthe signals that the source and other relays transmit exceeds .Thus,

for (59)

Defining and as the exponential orders of and ,respectively, we have

Choosing for a total of codewords, the fol-lowing expression for the conditional error probability can bederived:

Thus, is the set of channel realizations that satisfy

which can be simplified to

(60)

As before, is characterized by

(61)

Defining , , lets us simplify(60) and (61) to

(62)

(63)

From the definition of , it follows that

Note that (59) can also be simplified to

for


or

for (64)

In order to characterize , we need to consider threecases. The first case is when . In this case, (62) simplifiesto

Let us define , and. It immediately follows that , .

It is also easy to verify that

and (65)

where . From (65), it can be seen that

where

(66)The infimum value corresponds to and

or , , and , . Butwe assumed , so

(67)

From (64), it follows that

(68)

Now, from (66) and (68) we conclude that

(69)where

(70)

It turns out that (70) is an increasing function of . Therefore,its infimum corresponds to . Now, examining ,i.e.,

we realize that, for , it decreases with , thus, itsinfimum corresponds to . On the other hand, for

, becomes an increasing function of , whichmeans that its infimum corresponds to , i.e.,

.

(71)

The second case to be considered is when ,. It immediately follows that

(72)

In this case, (62) can be written as

(73)

If , then from (73), we get

(74)

On the other hand, from (64), it follows that

(75)Now, from (72), (74), and (75) one can see that

with

The infimum of corresponds to and, i.e.,

.(76)

If , then the problem of finding reduces

to the first case (i.e., ). Specifically, isgiven by (66), with , , , and substituting

, , , and . Thus,

where (77)

Note that (67) still holds. Derivation of fol-lows from (64)

with (78)


From (72), (77), and (78), we conclude that

(79)

where

(80)

As can be seen from (80), is a linear, and there-fore monotonic, function of . Thus, its infimum corresponds toeither or . Now if the infimum indeed corre-sponds to , by plugging in into (80), we derivea lower bound on it. That is,

or

for (81)

Choosing , on the other hand, gives

which has an infimum value, corresponding to andor , identical to the right-hand side of (81). This meansthat

for (82)

Now, from (82), (79), and (76), we conclude that

(83)

The third case (i.e., ), is trivial

(84)

From (71), (83), and (84) we conclude that (15) provides a lowerbound on the diversity gain achieved by the protocol. On theother hand, is also an upper bound on the diversity since:1) for , is the genie-aided diversity, 2) for

, it can be shown that the realization, where, , , and

corresponds to a channel outage for any , and 3) for, realization , , and

also corresponds to a channel outage for any . Thus, (15)is the diversity achieved by the relay DDF protocol andthe proof is complete.

F. Proof of Theorem 7

To characterize the diversity–multiplexing tradeoff achievedby the CB-DDF protocol, we first label the destinations ac-cording to the order in which they start transmission. That is,the first destination that starts transmission is denoted as desti-nation 1, the next destination as destination 2, and so on. Notethat the error probability of destination can be written as

(85)

where denotes the event that destination decodes the mes-sage and starts retransmission before the end of the codewordand is its complement. Now, since both and are lessthan one, (85) gives

(86)

In order to characterize , we need to characterize, i.e., destination ’s ML error probability, aver-

aged over the ensemble of Gaussian codebooks and conditionedon a certain channel realization, under the assumption that itstarted transmission before the end of the codeword. Towardthis end and through using Bayes’ rule, one can upper-bound

to get

(87)

Now, let us examine , i.e., the probability thatdestination , makes an error in decoding the sourcemessage, conditioned on (which ensures that destination

has indeed started retransmission) and assuming error-freedecoding at all of the active destinations. It follows fromLemma 1, that if the mutual information between the signalstransmitted by the source and active destinations and the signalreceived by destination exceeds (which is implied by ),then can be made arbitrarily small, provided

that the code length is sufficiently large. This means that forany and for sufficiently large code lengths

(88)

Using (88), (87) can be written as


This together with (86), yields

(89)

This means that the exponential order of provides alower bound on the diversity gain achieved by the protocol.Now, examining , it is easy to realize that the event in


which the th destination (out of destinations), spends theentire codeword listening, i.e., , is identical to the DDF relayprotocol with the rest of the destinations taking the role of the

relays. Thus, from (89), we see that communication tothe th destination achieves the same diversity order as doesthe DDF relay protocol with relays, namely, (18). Thiscompletes the proof.

G. Proof of Theorem 8

Realizing that (22) also corresponds to the optimal diversity–multiplexing tradeoff for a MIMO point-to-point communica-tion system with transmit antennas and a single receive an-tenna (i.e., the case of “genie-aided” cooperation betweensources), we only need to show that the CMA-NAF protocolachieves this tradeoff. To achieve this goal, we assume that eachof the sources uses a Gaussian random code with codewordlength and data rate , where is chosen as in (20) and growswith according to (21). We then characterize the joint ML de-coder’s error probability . Note that the error probabilityof the joint ML decoder upper-bounds the error probabilitiesof the source-specific ML decoders and thus provides a lowerbound on the achievable overall diversity gain (as a function of). In characterizing , we follow the approach of Tse et

al. [19] by partitioning the error event into the set of partialerror events , i.e.,

where denotes any nonempty subset of and(referred to as a “type- error”) is the event that the joint MLdecoder incorrectly decodes the messages from sources whoseindices belong to while correctly decoding all other messages.Because the partial error events are mutually exclusive

(90)

Using Bayes’ rule, one can upper-bound as

where, as before, and denote the outage event and itscomplement, respectively. The outage event is defined such that

dominates for all

(91)

Thus,

which, together with (90), results in

(92)

This means that , as defined by (91), provides an upperbound to the joint ML decoder’s error probability and there-fore a lower bound to the achievable diversity gain . Thederivation of , however, requires the characterization of

(i.e., the joint ML decoder’s type- PEP, condi-tioned on a particular channel realization and averaged over theensemble of Gaussian random codes). Here, we upper-bound

, for each , by the PEP of a suboptimal joint MLdecoder that uses only a subset of the destination’s observations(referred to as the type- decoder)

(93)

In (93), and represent the covariance matricescorresponding to the signal and noise components, respectively,of the partial observation vector used by the type- decoder,provided that the symbols of the sources that are not in set areset to zero. The size will be characterized in the sequel.

Before going into more detail on the type- decoder, we notethat, since and are both positive-definite matrices, theright-hand side of (93) can be upper-bounded as

(94)

The discussion is simplified if we define and as the ex-ponential orders of and , respectively. Note thatthe exponential orders of do not appear in the fol-lowing expressions for the reasons outlined in the proof of The-orem 3. We also note that the exponential orders of the broadcastgains are zero. Furthermore, recalling (5), the pdfsof negative and are effectively zero for large values of ,allowing us to concern ourselves only with their nonnegative re-alizations. With this ideas in mind, we return to (94) and claimthat

(95)

To understand (95), recall that the noise component of the des-tination observation is a linear combination of the noise origi-nating at the sources (i.e., ) and the noise originatingat the destination (i.e., ). Furthermore, the coefficients of thislinear combination are the products of some channel, broadcast,and repetition gains. Then, because these noise variances andmagnitude-squared gains can be written as nonpositive powersof , (95) must hold. Combining (95) and (94) yields

for (96)

As mentioned earlier, represents the covariance matrixof the signal component of the partial observation used by thetype- decoder, provided that the symbols of the sources that arenot in are set to zero. To fully characterize , though, wemust know which observations are used by the type- decoderand which are discarded. The type- decoder picks one obser-vation for every source in set , for a total of obser-vations per frame (where denotes the size of and therefore

). Provided that frame is not the last frame in itssuper-frame and assuming that during this super-frame, sourceis helping source , the destination observation componentcorresponding to source will be either the that correspondsto source ’s broadcast of or the that corresponds tohelper ’s rebroadcast of (where ). As an ex-ample, consider the case when and assume that during


(99)

(100)

a certain super-frame, source 3 is helping source (i.e.,, ). In this case, the type- decoder picks either

or in correspondence to . However, if instead ofsource 3, source 1 is helping source 2 (i.e., , ), thenthe type- decoder has to choose between or . Backto our description of the type- decoder, if , then the de-coder always picks over . On the other hand, if ,then the decoder chooses when or when

(i.e., the observation received through the betterchannel). The preceding discussion focused on the case whereframe is not the last frame of the super-frame. If frame isindeed last, then the decoder always chooses over .

We define , where , as the vector (of dimension) of contributions of symbol to the destination ob-

servations picked by the type- decoder. Clearly

Taking into account the independence of the transmitted sym-bols (i.e., ), we have

(97)

In order to illuminate some of the properties of , assume thatwe sort the chosen observations in chronological order. Fromthe description given, it is apparent that, associated with eachchosen observation (i.e., or ) there is one symbol(with ) which has contributions only from this observationforward. This means that if we define as

where and are chosen such that thefirst nonzero elements of , are sorted inchronological order, then will be lower-triangular and, con-sequently, will be upper-triangular. Furthermore, basedon the choice between or (corresponding to ), thefirst nonzero element of (i.e., the th diagonal elementof ) will be or , respectively. Next, wedefine as the signature of , i.e.,

and as

It follows then, that is also lower-triangular with the thdiagonal element being equal to or . Using thesedefinitions, (97) can be written as

(98)

The significance of can now be seen from the fact that (98)can be written as

Now, as the determinant of triangular matrices is simply theproduct of their diagonal elements, from (96) we get the firstequation at the top of the page. It is obvious that for large ’s,the previous inequality can be rewritten as (99) at the top of thepage. At rate and codeword length , and when thesymbols of the sources that are not in are set to zero, there area total of unique codewords. Thus, we have (100),also at the top of the page. This conditional type- error prob-ability leads to

where

(101)

Examining (101), we realize that for (91) to be met, shouldbe defined as the set of all real -tuples with nonnegativeelements that satisfy the following condition for at least onenonempty :

(102)


This way, by choosing large enough , can be made ar-bitrary large and thus (91) is always met. From (102), it followsthat

(103)

Substituting in this expression bygives

(104)

On the other hand, replacing in(103) by results in

(105)

Under the constraints given by (104) and (105), it is easy to seethat

(106)

Now, from (106) and (61), it follows that

Again, according to (92), provides a lower bound onthe diversity gain achieved by the protocol. Thus, the protocolachieves the diversity gain given by (22) and the proof iscomplete.

REFERENCES

[1] A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversity.Part I. System description,” IEEE Trans. Commun., vol. 51, no. 11, pp.1927–1938, Nov. 2003.

[2] , “User cooperation diversity. Part II. Implementation aspects andperformance analysis,” IEEE Trans. Commun., vol. 51, no. 11, pp.1939–1948, Nov. 2003.

[3] J. N. Laneman, D. N. C. Tse, and G. W. Wornell, “Cooperative diversityin wireless networks: Efficient protocols and outage behavior,” IEEETrans. Inf. Theory, submitted for publication.

[4] J. N. Laneman and G. W. Wornell, “Distributed space-time-coded pro-tocols for exploiting cooperative diversity in wireless networks,” IEEETrans. Inf. Theory, vol. 49, no. 10, pp. 2415–2425, Oct. 2003.

[5] A. Stefanov and E. Erkip, “Cooperative coding for wireless networks,”IEEE Trans. Commun., to be published.

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[7] R. U. Nabar, F. W. Kneubuhler, and H. Bölcskei, “Performance limitsof amplify-and-forward based fading relay channels,” in Proc. IEEE Int.Conf. Acoustics, Speech and Signal Processing, vol. 4, Montreal, QC,Canada, May 2004, pp. 565–568.

[8] R. U. Nabar, H. Bölcskei, and F. W. Kneubuhler, “Fading relay channels:Performance limits and space-time signal design,” IEEE J. Sel. AreasCommun., vol. 22, no. 6, pp. 1099–1109, Aug. 2004.

[9] N. Prasad and M. K. Varanasi, “Diversity and multiplexing tradeoffbounds for cooperative diversity protocols,” in Proc. IEEE Int. Symp.Information Theory, Chicago, IL, Jun./Jul. 2004, p. 268.

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[12] , “On the capacity of cooperative diversity in slow fading channels,”in Proc. 40th Allerton Conf. Communications, Control and Computing,Monticello, IL, Oct. 2002.

[13] , “A new achievable rate for cooperative diversity based on gen-eralized writing on dirty paper,” in Proc. IEEE Int. Symp. InformationTheory, Yokohama, Japan, Jun./Jul. 2003.

[14] M. A. Khojastepour, A. Sabharwal, and B. Aazhang, “Lower boundson the capacity of Gaussian relay channel,” in Proc. 38th Annu. Conf.Information Sciences and Systems, Princeton, NJ, Mar. 2004.

[15] , “On the capacity of Gaussian cheap relay channel,” in Proc IEEE2003 Global Communications Conf., San Francisco, CA, Dec. 2003.

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[17] S. Zahedi, M. Mohseni, and A. El Gamal, “On the capacity of AWGNrelay channel with linear relaying functions,” in Proc. IEEE Int. Symp.Information Theory, Chicago, IL, Jun. 2004, p. 399.

[18] L. Zheng and D. N. C. Tse, “Diversity and multiplexing: A fundamentaltradeoff in multiple antenna channels,” IEEE Trans. Inf. Theory, vol. 49,no. 5, pp. 1073–1096, May 2003.

[19] D. N. C. Tse, P. Viswanath, and L. Zheng, “Diversity–multiplexingtradeoff in multiple-access channels,” IEEE Trans. Inf. Theory, vol. 50,no. 9, pp. 1859–1874, Sep. 2004.

[20] K. A. Yazdi, H. El Gamal, and P. Schniter, “On the design of cooper-ative transmission schemes,” in Proc. 41st Allerton Conf. Communica-tion, Control, and Computing, Monticello, IL, Oct. 2003.

[21] K. Azarian, H. El Gamal, and P. Schniter, “On the achievable diver-sity-vs-multiplexing tradeoff in cooperative channels,” in Proc. Conf.Information Sciences and Systems, Princeton, NJ, Mar. 2004.

[22] H. El Gamal, “The diversity-multiplexing tradeoff in half-duplex co-operative channels: Achievable curves and optimal strategies,” in LIDSColloquia. Cambridge, MA: MIT Elec. Eng. Comp. Sci. Dept., May2004.

[23] K. Azarian, H. El Gamal, and P. Schniter, “On the achievable diversity-multiplexing tradeoff in half duplex cooperative channels,” in Proc. 42ndAllerton Conf. Communication, Control, and Computing, Monticello,IL, Oct. 2004.

[24] , “Achievable diversity-vs-multiplexing tradeoffs in half-duplex co-operative channels,” in Proc. IEEE Information Theory Workshop, SanAntonio, TX, Oct. 2004.

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tocOn the Achievable Diversity Multiplexing Tradeoff in Half-DuplexKambiz Azarian, Hesham El Gamal, Senior Member, IEEE, and PhilipI. I NTRODUCTIONII. B ACKGROUNDLemma 1: Consider a coherent linear Gaussian channel of data ratProof: Please refer to the Appendix .

III. T HE H ALF -D UPLEX R ELAY C HANNELA. Amplify and Forward (AF) ProtocolsTheorem 2: The optimal diversity gain for the cooperative relay Proof: Please refer to the Appendix .

Fig. 1. Optimal diversity multiplexing tradeoff for a single-relTheorem 3: The NAF protocol achieves the optimal diversity multiProof: Please refer to the Appendix .

Fig. 2. The super-frame in the NAF protocol with $N-1$ relays.Theorem 4: The diversity multiplexing tradeoff achieved by the NProof: The proof is virtually identical to that of Theorem 3, an

B. Decode and Forward (DF) Protocols

Fig. 3. Diversity multiplexing tradeoff for the DDF protocol witTheorem 5: The diversity multiplexing tradeoff achieved by the sProof: Please refer to the Appendix .

Fig. 4. Diversity multiplexing tradeoff for the NAF, DDF, LW-STCFig. 5. Diversity multiplexing tradeoff for the DDF protocol witTheorem 6: The diversity multiplexing tradeoff achieved by the DProof: Please refer to the Appendix .

IV. T HE H ALF -D UPLEX C OOPERATIVE B ROADCAST (CB) C HANNEL

Fig. 6. The cooperation frame, super-frame, and coherence intervTheorem 7: The diversity multiplexing tradeoff achieved by the CProof: Please refer to the Appendix .

V. T HE H ALF -D UPLEX C OOPERATIVE M ULTIPLE -A CCESS (CMA) C H

Fig. 7. Comparison of the outage probability for the NAF relay, Theorem 8: The CMA-NAF protocol achieves the optimal (genie-aideProof: Please refer to the Appendix .

VI. N UMERICAL R ESULTS

Fig. 8. Comparison of the outage probability for the DDF relay, Fig. 9. Comparison of the outage probability for the CMA-NAF, LTVII. C ONCLUSIONA. Proof of Lemma 1B. Proof of Theorem 2C. Proof of Theorem 3

Fig. 10. Outage region for the NAF protocol with a single relay.D. Proof of Theorem 5

Fig. 11. Outage region for the DDF protocol with a single relay Fig. 12. Outage region for the DDF protocol with a single relay E. Proof of Theorem 6F. Proof of Theorem 7G. Proof of Theorem 8A. Sendonaris, E. Erkip, and B. Aazhang, User cooperation diversJ. N. Laneman, D. N. C. Tse, and G. W. Wornell, Cooperative diveJ. N. Laneman and G. W. Wornell, Distributed space-time-coded prA. Stefanov and E. Erkip, Cooperative coding for wireless networM. Janani, A. Hedayat, T. Hunter, and A. Nosratinia, Coded coopeR. U. Nabar, F. W. Kneubuhler, and H. Bölcskei, Performance limiR. U. Nabar, H. Bölcskei, and F. W. Kneubuhler, Fading relay chaN. Prasad and M. K. Varanasi, Diversity and multiplexing tradeofG. Kramer, M. Gastpar, and P. Gupta, Cooperative strategies and A. Host-Madsen, On the capacity of wireless relaying, in Proc. 2M. A. Khojastepour, A. Sabharwal, and B. Aazhang, Lower bounds oU. Mitra and A. Sabharwal, On achievable rates for complexity coS. Zahedi, M. Mohseni, and A. El Gamal, On the capacity of AWGN L. Zheng and D. N. C. Tse, Diversity and multiplexing: A fundameD. N. C. Tse, P. Viswanath, and L. Zheng, Diversity multiplexingK. A. Yazdi, H. El Gamal, and P. Schniter, On the design of coopK. Azarian, H. El Gamal, and P. Schniter, On the achievable diveH. El Gamal, The diversity-multiplexing tradeoff in half-duplex K. Azarian, H. El Gamal, and P. Schniter, On the achievable diveT. M. Cover and A. A. El Gamal, Capacity theorems for the relay T. M. Cover and J. A. Thomas, Elements of Information Theory: Wi

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